Threshold Pion Production at Large Momentum Transfers V. M. BRAUN

Transcription

1 Threshold Pion Production at Large Momentum Transfers V. M. BRAU University of Regensburg based on V.M. Braun, D. Ivanov, A. Lenz and A. Peters, Phys.Rev.D75:040,007 V.M. Braun, D. Ivanov and A. Peters, Phys.Rev.D77:03406,008 PAIC 008, Eilat, 3..08

2 Electroproduction with Q in a few GeV range: Tradition: excitation of nucleon resonances (transition form factors) e(l) + p(p) e(l ) + (3)(P ) e(l) + p(p) e(l ) + (440)(P ) Proposal: pion electroproduction close to threshold W W th e(l) + p(p) e(l ) + π + (k) + n(p ) e(l) + p(p) e(l ) + π 0 (k) + p(p ) W = (P + k) W th = m + m π Q = = (l l ) Hard (pqcd) and soft (ChPT) physics meet together!

3 Generalized Form Factors = S-wave Multipoles at Threshold at the threshold ( π j em µ p = i (P )γ 5 γ µ µ f π m G π (Q ) iσµνν m G π (Q ) ) (P ) related to S-wave multipoles in the PWA, e.g. for m π = 0 E π 0+ (Q, W th) = 4παem Q Q + 4m 8π m 3 G π fπ L π 0+ (Q, W th) = 4παem Q Q + 4m 3π m 3 G π fπ e.g. the differential cross section at threshold is given by dσ γ = dω π th k f W h W m (E π 0+ ) + ǫ Q (ωγ th (Lπ 0+ )i )

4 Chiral rotation Spontaneous Breaking of Chiral Symmetry In the chiral limit, m π/m 0, the pion can be rotated away: p = φs(x) 6 u d u u u d d u u + φa(x) u u d d u u p π 0 = n π + = φ s(x) 6u d u + u u d + d u u φa(x) 6f π u u d d u u f π φ s(x) u d u 3u u d 3d u u φa(x) u u d d u u fπ f π Pobylitsa, Polyakov, Strikman; PRL87(00)000 allows one to look at the proton from a different angle The relevant degrees of freedom change with Q rich physics rich theory

5 Q < 0. GeV : Chiral Perturbation Theory local effective low-energy theory systematic expansion in powers of m π and applicable for m π < 300 MeV(?) Bernard, Kaiser, Meissner; IJMP, E4 (995)93 Drechsel, Tiator; J. Phys. G 8 (99) 449 Kroll, Ruderman µ = m π/m /7 E π+ n 0+ ( = 0, W th ) = eg» π 3 8πm µ + O(µ ln µ ) = /m π exp: 7.9 ± 0.5; 8.8 ± 0.7 «#) m ( " E π0 p 0+ ( = 0, W th ) = eg π µ µ 8πm (3 + κp) + 4f π Subtlety: 0 and m π 0 limits do not commute

6 Chiral Perturbation Theory continued ambu, Lurié, Shrauner E ( ) 0+ (mπ = 0, ) = eg A 8πf π ( + 6 r A + 4m «G A ( ) = g A + 6 r A +... Bernard, Kaiser, Meissner; PRL69 (99)877 κ v + «+ 8fπ «) π Experiment: r A = 0.65 ± 0.03 (elastic ep); r A = ± 0.05 (pion el.prod) S-wave cross section γ p π 0 p W = 074, ǫ = 0.58 Q

7 Q m 3 /m π: Low-Energy Theorems predate ChPT and QCD Chiral symmetry: pion mass m π 0 pion coupling k 0 Pion emission from external legs P P P P P P k k k Chiral Rotation c P P π a j em µ i f π [j em µ, Qa 5 ] k Kroll, Ruderman 54

8 Low-Energy Theorems continued PCAC + current algebra: Q m Q m G π0 p = g A Vainshtein, Zakharov, PB36(97)589 Scherer, Koch, PA534(99)46 Q (Q + m ) Gp M, 0 p Gπ = g Am (Q + m ) Gp E, G π+ n = g A Q (Q + m ) Gn M + G A, G π+ n = g A m (Q + m ) Gn E, Derivation does not imply Q m π! Threshold photoproduction of π 0 is suppressed compared to π + The π 0 /π + ratio is rapidly increasing with Q The O(m π) corrections can be added but, no systematic way to treat O(m π ) terms (ChPT)

9 Low-Energy Theorems continued (II) expected to fail for Q m3 m π since π cannot have small momentum w.r.t. the initial and final state protons simultaneously P P P k P P k k at threshold m (P k) = mπ h i Q + m m phenomenological Lagrangians to take into account nucleon resonances or go over to uark-gluon description

10 Q m 3 /m π: Perturbative QCD QCD factorization for Q Λ 3 QCD /mπ Pobylitsa, Polyakov, Strikman, PRL87(00)000: P P k soft hard soft p = φs(x) 6 u d u u u d d u u + φa(x) u u d d u u p π 0 = n π + = φ s(x) 6u d u + u u d + d u u φa(x) 6f π u u d d u u f π φ s(x) u d u 3u u d 3d u u φa(x) u u d d u u fπ f π Only for G π (E 0+) Probably unrealistic at reachable momentum transfers

11 Q m 3 /m π: Light Cone Sum Rules: Q 0 GeV normalized to the dipole formula G D = /( + Q /0.7) L π0 p 0+ /G D E π0 p 0+ /G D E π+ n 0+ /G D L π+ n 0+ /G D green dots: MAID; : W = 074 MeV, : W = 084 MeV, : W = 094 MeV solid curves: LCSRs using experimental elastic EM formfactors as input dashed curves: pure LCSRs, no experimental input

12 Light Cone Sum Rules continued Deviation from LET: (G π 0 p π ) QCD (G 0 p (G π 0 p ) QCD ) LET (G π 0 p 3 ) LET Q, GeV Q, GeV Reproduce LET for Q GeV Reproduce pqcd for Q (part of the LO α s contribution) o double counting of soft and hard contributions Tested: Electromagnetic and axial form factors, heavy meson decays, pion form factors

13 Moving away from threshold Higher partial waves P-wave dominated by pion emission from the final state P P k Energy dependence E 0+(W), L 0+(W), etc. due to final state interactions P k P = P k P

14 SLAC E36 P. E. Bosted et al.; PRD49(994) F p (W, Q ) F p (W, Q ) W, GeV W, GeV Figure: The structure function F p (W, Q ) as a function of W scaled by a factor 0 3 compared to the SLAC E36 data at the average value Q = 7.4 GeV (left panel) and Q = 9.43 GeV (right panel).

15 CLAS (preliminary) ep eπ + n MAID07 MAID07 Kijun Park; DP 008 Oakland, CA (Oct. 3-6) MAID07

16 Summary Use pion as a handle to rotate the nucleon wave function a novel object: generalized form factor; an overlap between usual and rotated WF check Low Energy Theorems (ambu,... ) and transition to QCD new scale in QCD: Q m 3 /mπ measure nucleon axial form factor (reuires π + ) theory progress feasible, large community (ChPT, pqcd, PWA) an (almost) untouched terrain...! no data at Q 0. GeV MAMI? Perfectly suited for the JLab GeV upgrade physics program

17 Supplementary Material

18 Q 0 GeV : Light Cone Sum Rules consider Z Tν π (P, ) = i Balitsky, V.B., Kolesnichenko d 4 x e ix (P )π a (k) T{j em ν (x) ηp(0)} 0 h i η p(x) = ǫ ijk u i (x)cγ µu j (x) γ 5γ µ d k (x), 0 η p (P) = λ pm (P) take P = P + k, P GeV and make a matching between (a) The Operator Product Expansion in terms of pion-nucleon DAs (P )π a (k) T{j em ν (x) ηp(0)} 0 = X H ν(x, px) 0 (x )(x )(x 3) (P )π a (k) twist (b) The dispersion representation in terms of hadronic states P π P P P Borel transformation to improve convergence

19 Light Cone Sum Rules continued Good things: Reproduce LET for Q GeV Reproduce pqcd for Q (part of the LO α s contribution) o double counting of soft and hard contributions Tested: Electromagnetic and axial form factors, heavy meson decays, pion form factors ot so good things: Use nucleon distribution amplitudes as input not so well known Calculation rather demanding, especially in LO Bad things: Approximation for the continuum contribution not improvable irreducible error of order 0% for all Q

20 Light Cone Sum Rules: ucleon Electromagnetic form factors G M / GD G E / GM choice of nucleon DAs: solid: BLW model long dashes: asymptotic short dashes: CZ model Braun, Lenz, Wittmann; PRD73(006)09409 A. Lenz; arxiv: v

21 Light Cone Sum Rules: Pion ucleon Intermediate States ew: Semidisconnected pion-nucleon contributions in the intermediate state P P P a) k k b) k c) Figure: Schematic structure of the pole terms in the correlation function b) and c) correspond to π coupling to the Ioffe current 0 η (P» p(0) k)π(k) = iλp m g A f π P m ( P k + m ) k γ 5(P k). In the threshold kinematics, with δ = m π/m 8 T π0 p ν (P, ) = iλp m < (( + δ) P + m )γ 5 f π : m P 4(γ ν ν ) Gπ0 p m + ( + δ)γ 5( P + m ) p iσνµµ»γ νf [m ( + F p δ) + δq ] P m»(γ ν ν )G ( + δ)ga( P + m)γ5 [m ( + δ) + δq ] P iσνµµ m pm iσνµµ 4m m Gp E 3 G π 0 p 5 ff (P)

22 Light Cone Sum Rules: Pion ucleon Intermediate States cont. The semidisconnected π contributions can be included in the continuum if m πq > m (s 0 m ) Q > 7 GeV [ Λ 3 QCD /mπ] Otherwise they have to be taken into account explicitly Q m G π0 p G π0 p Q m G π+ n = em /M " λ p B P [A π0 p ](M, Q ) e δ(m +Q )/M F p (Q ) g AQ # Q + m G p M (Q ) = em /M " λ p B P [B π0 p δ(m ](M, Q ) + e +Q )/M Fp (Q ) + g Am # Q + m G p E (Q ) /M " λ p B P [A π+ n ](M, Q ) e δ(m +Q )/M F n (Q ) g AQ # Q + m G n M (Q ) " B P [B π+ n δ(m ](M, Q ) + e +Q )/M F n (Q ) + g A m # Q + m G n E (Q ) = em G π+ n = em /M λ p where A(P, Q ) and A(P, Q ) are the invariant functions defined as z ν Λ + T π ν (P, ) = i n (pz + kz)γ 5 m A(P, Q )+ B(P, Q o ) + (P) f π

23 π scattering phases Figure from ozawa et al., PRC4(990)3

24 Including P-waves: for W W th m π accept (P )π(k) j em µ (0) p(p) = = i j (P )γ 5 γ µ µ f π m G π (Q ) iσµνν m j ic πg A + f π[(p +k) ) m ] (P ) k γ 5( P +m ) F p (Q ) γ µ µ ff G π (Q ) (P) «ff + iσµνν F p m (Q ) (P) S wave: generalized form factors from LCSR P wave: pion emission from the final state nucleon; exact in chiral limit Eventually can take into account the final state interactions G π (Q ) G π (Q, W) G π (Q )[ + i t π]

25 Structure Functions at x B F (W, Q ) = β(w) (4πf π) F (W, Q ) = β(w) (4πf π) g (W, Q ) = β(w) (4πf π) X ( Q + 4m π 0,π + m 4 ( X Q π 0,π + m 4 ( X Q π 0,π + m 4 g (W, Q ) = β(w) ( X Q (4πf π) π 0,π + m 4 Q G π + c π g A W β ) (W) 8(W m Q m ) G M! Q G π + m 4 Q G π + c π g A W β (W)Q m Q 4(W m ) h Q G π m Re(Q G π G,π h Q G π + 4 Q Re(Q G π G,π G M +4m!) G E Q +4m i ) + c π g A W β ) (W) 8(W m Q m ) G MF p i ) + c π g A W β ) (W) 3(W m Q4 G M F p ) β(w) = k f W, xb = Q Q + W m

26 Differential Cross Section For unpolarized protons, the virtual photon cross section is with M γ = M T + ǫ M L + dσ γ = αem 8π k f W dω π W m M γ ǫ( + ǫ) M LT cos(φ π) + ǫm TT cos(φ π) + λ ǫ( ǫ) M LT sin(φπ) f π MT = 4 k i Q m G π + c π g Ak f (W m Q m ) G M f π ML = k i Gπ + 4c π g A k f (W m ) m4 G E f π cπga ki kf cos θ W m cπga ki kf h i MLT = sin θ W m Qm G MRe G π + 4G ERe G π f π MTT = 0, f π M LT = sin θ cπga ki kf h i W m Qm G MIm G π 4G EIm G π cπga ki kf + cos θ W m 4Q G MRe G π 4m GERe Gπ M TT = 0: no D wave; tests uality of the approximation M LT : single spin asymmetry; arises because of FSI, calculable

27 Miscellaneous Results F γ p π 0 p /F γ p X W, GeV dσ γ p π 0 p /dω π, µb/ster cos θ Figure: The fraction of π 0 p in F p (W, Q ) for Q = 3 GeV (upper curve) and Q = 9 GeV (lower curve) Figure: Differential cross section dσ γ p π 0 p /dω π for φ π = 35 grad, Q = 4. GeV and W =. GeV p 0 F (W, Q ) W, GeV Figure: S-wave (solid) vs. P-wave (dashed) for F p (W, Q ) at Q = 7.4 GeV Q 6 σ γ p π 0 p Q, GeV Figure: Integrated cross section Q 6 σ γ p π 0 p for W =. GeV (lower curve) and W =.5 GeV (upper curve)